4B15 Introduction to Bioengineering
Electrodes and Transducers
Lecture 5: Non-Linearity in Transducers
5.1 Introduction
Apart from nominal manufacturing variations and temperature effects on
the performance of the transducer, there are other factors which cause errors in
the measurement and limit the precision which can be obtained. Transducers
rarely possess a perfectly linear transfer characteristic, but always have some
degree of non-linearity over their range of operation and deviate from the ideal
straight-line output voltage vs input pressure relationship as shown in Fig. 1.
Second order non-linearity in the characteristic usually limits the error to one
direction: either pressure will be slightly underestimated or slightly
overestimated, depending on the slope of the curve. Third order non-linearity
means that the slope of the characteristic can change direction across the range
of operation so that the pressure may be underestimated in one part of the range
and overestimated in another part.
Moreover properties such as hysteresis and ageing effects can also give
rise to measurement errors. Hysteresis means that the transducer may give a
slightly different output voltage for the same value of applied pressure
depending on whether the pressure is increasing or decreasing. These sources of
error are usually quantified by manufacturers in the data sheets for the devices
as shown in Table 1 below.
output
voltage
2nd order non-linearity
sensitivity
variation
offset
variation
ideal linear characteristic
3rd order non-linearity
applied pressure
Fig. 1 Transfer Characteristic of the SX 50 Pressure Transducer
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Table I
The Performance Parameters of the SX 50 Pressure Transducer
Parameter
Min.
Typ.
Max.
Unit
Pressure Range:
0
-
100
kPa
Supply Current:
-
2.7
-
mA
Full-scale Span:
75
110
150
mV
Sensitivity:
750
1100
1500
µV/kPa
Offset Voltage:
-35
-20
0
mV
-
+48
-
µV/OC
-2400
-2150
-1900
ppm/OC
-
4.65
-
kΩ
+690
+750
+810
ppm/OC
-
±0.2
±0.5
%FS
Offset T. C.:
Sensitivity T.C.:
Bridge Resistance:
Resistance T.C.:
Lin & Hys Error:
5.2 Non-linearity: Static Analysis
A transfer characteristic having second order non-linearity can be
described by a quadratic in the applied pressure as:
where the coefficients are generic and describe the degree of non-linearity
present. If a two-point calibration method is used then the conditioning
amplifier will be adjusted to give and output of 0V for an input applied pressure
of zero. If this is the case then the coefficient ‘a’ is effectively adjusted to zero so
that the amplifier output becomes:
The coefficient ‘c’ will normally be much smaller than ‘b’ as only a small degree
of non-linearity should be present. Ideally the transducer output should be
purely linear and determined by the bride supply voltage and the span
sensitivity of the transducer such that:
The error in the output voltage can then be described by the difference between
the actual and the ideal characteristics:
2
If a two-point calibration procedure is used so that the amplifier gain or the
bridge supply voltage is adjusted to give the required maximum output voltage
at full-scale pressure input, pmax. Then the error can be taken as zero for
maximum input pressure so that:
so that:
The error can then be described by:
The pressure at which maximum error occurs can be found by taking the
derivative as:
which gives:
So that:
This shows that the maximum error caused by non-linearity occurs at half fullscale. The value of this error is given as:
So finally:
3
Whether this error is positive or negative depends on whether the coefficient ‘b’
is greater than or less than SVB.
This error is the worst-case value for 2nd order non-linearity. It can
therefore be used to indicate the maximum error caused by non-linearity in the
transducer. This is done by expressing its magnitude as a fraction or percentage
of the ideal full scale output voltage. This gives:
5.3
!% $
#! !%
Non-linearity: Simple Dynamic Input
The problem with non-linearity is evident when a time varying input is
involved. Thus, when monitoring a pressure which varies with time the amplifier
will generate an electrical output signal which is a function of time. This is
subject to the nonlinearity of the transducer characteristic so that:
If, for example, the input pressure is taken as being sinusoidal in nature as
is described as:
%&
Then the output voltage is given as:
%& % &
Using the trigonometric identity cos2A= 1/2[1+cos(2ωt)] gives:
%& . (
. (
%&
Note that the first term in this expression is a scaled, presumably amplified,
version of the input pressure. The second term represents a dc component which
is generated at the output but is not present at the input. This is essentially an
offset voltage which is generated solely as a result of the non-linearity of the
transducer. The third term is a term with a time profile at twice the frequency of
the input signal, that is, it is a second harmonic of the fundamental component
which has a radian frequency ω. This leads to second harmonic distortion of the
profile of the original pressure signal when observed at the output of the
conditioning amplifier.
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4.3 Non-linearity: Composite Input
Consider a second input pressure scenario where the input consists of two
frequency components
! %&! %& Then in this case the output voltage from the amplifier will be described by:
)! %&! %& * )! %&! %& *
This expands to:
! %&! %& ! % &! % & ! %&! %& ! %&! %& . (
! . (
! %&! . (
. (
%& ! %&! %& Using the identity 2cosAcosB = cos(A+B) + cos(A-B) gives:
! %&! %& . (
! . (
. (
! %&! . (
%& ! %&! & ! %&! & The terms on the left hand side on the first line are the scaled input components.
The terms on the right hand side of the first line are the dc offset components,
with an offset contributed by each component of the input pressure.
The terms on the second line are the second harmonic components of each of the
input frequencies present.
Finally the terms on the last line are the sum and difference components of the
input frequencies. These are referred to as intermodulation components as they
are not harmonically related to the frequencies present in the input applied
pressure profile. Because of the lack of harmonic quality they often result in
more serious distortion than that caused by the harmonic components. In
speech, for example, they would have a more detrimental effect on intelligibility
than the harmonic components.
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Consider ω1 = 100Hz and ω2 = 80Hz
Then ω1 + ω2 = 180Hz
and ω1 - ω2 = 20Hz
The sum component at 180Hz is higher than the highest input frequency
and so could perhaps be filtered out using a low-pass filter.
The difference component is lower than the lowest input frequency and hence
cannot be removed by low-pass filtering. High-pass filtering may appear to be
an option but this would not be appropriate if there were a dc or steady-state
component in the input applied pressure. The difference component is at a very
low frequency and this can cause a low drift or wander-like modulation of the
output signal. Generally little can be done in analogue circuitry to counteract the
effects of transducer non-linearity. It is usually a matter of deciding on the
degree of non-linearity error that can be accepted and selecting a suitable
transducer accordingly.
With more expense, analogue-to-digital conversion can be used in
conjunction with a look-up table to correct non-linearity, usually for the
nominal operating temperature. The look-up table stores a corrected value of
the transducer reading which is accessed based on its actual reading and a
previous measure of the non-linearity of its transfer characteristic.
4.3 Non-linearity: Composite Input
An example of a pressure monitor used for blood pressure measurement is
shown in Fig.2. This circuit includes full temperature compensation of both
offset and span sensitivity. It has a two-point calibration which can be carried
out in a single iteration of the temperature cycle. It provides a digital readout of
pressure but does not include non-linearity correction.
The circuit is powered from a battery which feeds a 5V regulator, IC1,
with a low dropout voltage. The buffer op-amp IC6 is used to generate a -0.5V
secondary supply voltage to feed the other op-amps IC4 and IC5 which cannot
operate from ±5V. The heart of the electronic manometer is the digital panel
meter (DPM), IC7, a MAX138 (Maxim Inc.), which is a differential-input,
ratiometric DPM, having a very high input common-mode-rejection-ratio, high
temperature-stable performance, as well as an on-chip positive-to-negative
supply converter. It provides a direct drive facility for a 3½ digit liquid crystal
display (LCD). The numerical output displayed on the LCD is given as:
!
+,+ +./,+ ./-
The reference voltage is obtained via a potential divider, R16, R17 and RV5, placed
across an internal precision voltage reference which exists between the supply
rail and the analogue common pin on the DPM, IC7 pin 32. The potentiometer
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Fig. 2 Schematic Diagram of Pressure Transducer for Blood Pressure Monitor
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RV5 is used to adjust the nominal full-scale value on the display at the lower
calibration temperature. The other ancillary components connected to IC7 are
selected in accordance with the manufacturer's recommendations for optimum
performance.
The SX15 transducer, IC2, is driven by a programmable current source,
IC3, the value of current being set by resistors R2, R3 and the voltage at the
output pin 1 of op-amp IC4A. These components are also used to match the
temperature coefficient of the current source to that of the transducer in order
to provide nominal temperature compensation of the full-scale span over the
working range. IC4 is chosen to have a very low temperature coefficient of offset
voltage so that the voltage at pin 1 is essentially temperature-independent. The
input potential to IC4A, which acts as a buffer amplifier, is provided by the
potential divider consisting of the bandgap reference diode, ZD1, having a very
low temperature coefficient, resistors R4 – R7 and RV1 which is used to adjust it.
The positive output voltage from pin 2 of the transducer, IC2, is fed via one side
of the low-pass filter comprising of R15 and C8 to the positive differential input
pin, IN HI, of the panel meter, IC7 pin 31. The output voltage from the negative
side of the transducer, IC2 pin 4, is scaled by a factor of ½ by the potential
divider consisting of R8, R9 and RV2, but is subsequently given a gain of 2 in the
summing amplifier composed of IC5B, R12 and R13. The potentiometer RV2 is
used to cancel the nominal offset voltage of the transducer at the lower
calibration temperature. The output of the amplifier, IC5B pin 7, is fed into the
negative differential input of the panel meter, IN LO, IC7 pin 30, via the other
side of the low-pass filter composed of R14 and C8. This ensures that only slowly
changing pressure is measured and displayed on the LCD.
The output voltage at pin 3 of the programmable current source, IC3, has
precise and linear temperature dependence but is not zero at room temperature.
The resistors R2 and R3 are used to modify the effective temperature coefficient
of the current source by the ratio of their values so that it closely matches the
nominal value of the transducer temperature coefficient. The negative potential
at the output of IC4A is then used to obtain a zero potential at node A at the
lower calibration temperature, which then increases linearly above this
temperature. This potential is fed to one side of the two potentiometers, RV3 and
RV4 and also to the inverting amplifier consisting of IC4B and resistors R10 and
R11 to generate an equal negative temperature dependent voltage which is then
fed to the other side of the potentiometers. With IC4 chosen for a very low
temperature coefficient of offset voltage, this provides perfectly balanced
positive and negative temperature dependent voltages on each side of the
potentiometers. This mechanism also means that, regardless of the setting of RV3
and RV4, the potentials at the wipers of both of these potentiometers are zero at
the lower calibration temperature as required for a single cycle calibration
procedure. The potential at the output of RV4 is added to the differential voltage
of the transducer in the amplifier consisting of IC5B and resistors R12 and R13.
This is then adjusted to correct the temperature coefficient of the offset voltage
of the transducer at the upper calibration temperature. The potential at the
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output of RV3 is added to the nominal reference voltage of the DPM in the
amplifier IC5A and resistors R18 – R21. This is adjusted to counteract the
temperature coefficient of the transducer sensitivity at the upper calibration
temperature. The following calibration procedure is used:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
All potentiometers are centred initially.
The oven temperature is brought to 20O C.
RV1 is adjusted to give 0V at node A.
An input pressure of 0mmHg is applied.
RV2 is adjusted until the LCD displays ''000''.
The input pressure is increased to 300mmHg.
RV5 is adjusted until the LCD reads ''300''.
The oven temperature is then raised to 40O C.
The input pressure is set to 0mmHg again.
RV4 is adjusted until the LCD reading is ''000''.
The input pressure is raised to 300mmHg.
RV3 is adjusted until the LCD reading is ''300''.
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